Piddock was born in Rochester, Kent, the son of Celia May (née O'Callaghan) and Charles Frederick Piddock. He made his theatrical debut in the U.S. in The Boy's Own Story, a one-man show about a soccer goalkeeper, at the Julian Theatre in San Francisco. The show won Piddock the Bay Area Critics' Best Actor Award. The show was then produced Off Broadway.

Piddock appeared in his first movie in 1989, Lethal Weapon 2 as the South African consulate. Other film roles soon followed, including Independence Day, Traces of Red, Multiplicity, Burn Hollywood Burn, Austin Powers in Goldmember, A Different Loyalty, Love for Rent, See This Movie, The Prestige, Epic Movie, Who's Your Caddy?, Meet The Spartans and The Seeker: The Dark Is Rising. He also voiced Zazu in The Lion King: Simba's Mighty Adventure and Timon and Pumbaa's Jungle Games. He also provided the voice for Alfred Pennyworth in Batman: Under the Red Hood, as well as Agent One in Return to Castle Wolfenstein for the PlayStation 2 and Xbox.

When asked on his Twitter account if he was going to reprise the role of Zero for Metal Gear Solid V: The Phantom Pain, a week after Banjō Ginga was confirmed to be a member of the Japanese cast, Piddock replied on two different posts with: "Unlikely, unless they increase their derisory offer to slightly less derisory!" and "Ha! Thanks. Tell the makers of the game not to make Major Zero such an insulting offer and he'll be there!"[1] Piddock later elaborated on the reason for the refusal in an episode of The Codec, explaining that, because Zero has a limited physical presence in the game, Konami felt justified in giving him what he believed was too small of a payment for his voice work for the character, although he implied that Zero would be mostly heard instead of seen.[2]